Having lost to Wolfsburg and Bayern Munich in their last two league outings, Pierre-Emerick Aubameyang gave Dortmund the lead after 70 minutes as he curled home a superb free-kick.
However, Mainz were level within four minutes when Eric Maxim Choupo-Moting converted a penalty, after Lukasz Piszczek was penalised for a shove on Yunus Malli in the box.
The defender was given a straight red for his actions before Lewandowski tucked away the third penalty of the match in injury time to secure Dortmund's victory.
Piszczek made his first start for Dortmund since May's UEFA Champions League final, with Kevin Grosskreutz moving into midfield alongside Sven Bender, who started despite breaking his nose against Napoli on Tuesday.
Aubameyang replaced Henrikh Mkhitaryan in Dortmund's forward line.
Zdenek Pospech was named in the Mainz starting XI, meanwhile, despite having been taken off in last Sunday's 3-2 win over Werder Bremen, while Shinji Okazaki led the line having scored four times in his last four Bundesliga games.
The visitors' first effort on goal came through Lewandowski after 25 minutes, with the Poland international expertly controlling a Marco Reus cross before miscuing his effort and firing wide.
Mainz defender Stefan Bell first saw his header hit the woodwork from a corner, before Aubameyang's dipping strike was tipped onto the frame by Loris Karius.
Weidenfeller was then called on to make a great save from Choupo-Moting's fierce drive two minutes into the second half before Okazaki fired over from the left-hand side of the area.
It was Dortmund who took the lead 20 minutes from time, though, with Aubameyang's expertly-hit free-kick putting the visitors ahead.
Jurgen Klopp's side hit back almost immediately when Lewandowski converted from 12 yards following Soto's cynical handball to deny Aubameyang a second.
